GALAHAD CQP package#

purpose#

The cqp package uses a primal-dual interior-point method to solve a given convex quadratic program. The aim is to minimize the quadratic objective function

\[q(x) = f + g^T x + \frac{1}{2} x^T H x,\]

or the shifted-least-distance objective function

\[s(x) = f + g^T x + \frac{1}{2} \sum_{j=1}^n w_j^2 (x_j - x_j^0)^2,\]

subject to the general linear constraints and simple bounds

\[c_l \leq A x \leq c_u \;\;\mbox{and} \;\; x_l \leq x \leq x_u,\]

where $H$ and $A$ are, respectively, given $n$ by $n$ symmetric postive-semi-definite and $m$ by $n$ matrices, $g$, $w$ and $x^0$ are vectors, $f$ is a scalar, and any of the components of the vectors $c_l$, $c_u$, $x_l$ or $x_u$ may be infinite. The method offers the choice of direct and iterative solution of the key regularization subproblems, and is most suitable for problems involving a large number of unknowns $x$.

See Section 4 of $GALAHAD/doc/cqp.pdf for additional details.

terminology#

Any required solution $x$ necessarily satisfies the primal optimality conditions

\[A x = c\;\;\mbox{(1a)}\]

and

\[c_l \leq c \leq c_u, \;\; x_l \leq x \leq x_u,\;\;\mbox{(1b)}\]

the dual optimality conditions

\[H x + g = A^{T} y + z,\;\; y = y_l + y_u \;\;\mbox{and}\;\; z = z_l + z_u,\;\;\mbox{(2a)}\]

and

\[y_l \geq 0, \;\; y_u \leq 0, \;\; z_l \geq 0 \;\;\mbox{and}\;\; z_u \leq 0,\;\;\mbox{(2b)}\]

and the complementary slackness conditions

\[( A x - c_l )^{T} y_l = 0,\;\; ( A x - c_u )^{T} y_u = 0,\;\; (x -x_l )^{T} z_l = 0 \;\;\mbox{and}\;\;(x -x_u )^{T} z_u = 0,\;\;\mbox{(3)}\]

where the vectors $y$ and $z$ are known as the Lagrange multipliers for the general linear constraints, and the dual variables for the bounds, respectively, and where the vector inequalities hold component-wise.

In the shifted-least-distance case, $g$ is shifted by $-W^2 x^0$, and $H = W^2$, where $W$ is the diagonal matrix whose entries are the $w_j$.

method#

Primal-dual interior point methods iterate towards a point that satisfies these optimality conditions by ultimately aiming to satisfy (1a), (2a) and (3), while ensuring that (1b) and (2b) are satisfied as strict inequalities at each stage. Appropriate norms of the amounts by which (1a), (2a) and (3) fail to be satisfied are known as the primal and dual infeasibility, and the violation of complementary slackness, respectively. The fact that (1b) and (2b) are satisfied as strict inequalities gives such methods their other title, namely interior-point methods.

The method aims at each stage to reduce the overall violation of (1a), (2a) and (3), rather than reducing each of the terms individually. Given an estimate $v = (x, \; c, \; y, \; y^{l}, \; y^{u}, \; z, \; z^{l}, \; z^{u})$ of the primal-dual variables, a correction $\Delta v = \Delta (x, \; c, \; y, \; y^{l}, \; y^{u} ,\;z,\;z^{l} ,\;z^{u} )$ is obtained by solving a suitable linear system of Newton equations for the nonlinear systems (1a), (2a) and a parameterized ``residual trajectory’’ perturbation of (3); residual trajectories proposed by Zhang (1994) and Zhao and Sun (1999) are possibilities. An improved estimate $v + \alpha \Delta v$ is then used, where the step-size $\alpha$ is chosen as close to 1.0 as possible while ensuring both that (1b) and (2b) continue to hold and that the individual components which make up the complementary slackness (3) do not deviate too significantly from their average value. The parameter that controls the perturbation of (3) is ultimately driven to zero.

If the algorithm believes that it is close to the solution, it may take a speculative ``pounce’’ extrapolation, based on an estimate of the ultimate active set, to avoid further costly iterations. If the pounce is unsuccessful, the iteration continues, but further pounces may be attempted later.

The Newton equations are solved by applying the matrix factorization package SBLS, but there are options to factorize the matrix as a whole (the so-called “augmented system” approach), to perform a block elimination first (the “Schur-complement” approach), or to let the method itself decide which of the two previous options is more appropriate. The “Schur-complement” approach is usually to be preferred when all the weights are nonzero or when every variable is bounded (at least one side), but may be inefficient if any of the columns of $A$ is too dense.

Optionally, the problem may be pre-processed temporarily to eliminate dependent constraints using the package FDC. This may improve the performance of the subsequent iteration.

references#

The basic algorithm is a generalisation of those of

Y. Zhang, ``On the convergence of a class of infeasible interior-point methods for the horizontal linear complementarity problem’’. SIAM J. Optimization 4(1) (1994) 208-227,

and

G. Zhao and J. Sun, ``On the rate of local convergence of high-order infeasible path-following algorithms for $P_*$ linear complementarity problems’’. Computational Optimization and Applications **14(1)* (1999) 293-307,

with many enhancements described by

N. I. M. Gould, D. Orban and D. P. Robinson, ``Trajectory-following methods for large-scale degenerate convex quadratic programming’’, Mathematical Programming Computation 5(2) (2013) 113-142.

matrix storage#

unsymmetric storage#

The unsymmetric $m$ by $n$ matrix $A$ may be presented and stored in a variety of convenient input formats.

Dense storage format: The matrix $A$ is stored as a compact dense matrix by rows, that is, the values of the entries of each row in turn are stored in order within an appropriate real one-dimensional array. In this case, component $n \ast i + j$ of the storage array A_val will hold the value $A_{ij}$ for $1 \leq i \leq m$, $1 \leq j \leq n$. The string A_type = ‘dense’ should be specified.

Dense by columns storage format: The matrix $A$ is stored as a compact dense matrix by columns, that is, the values of the entries of each column in turn are stored in order within an appropriate real one-dimensional array. In this case, component $m \ast j + i$ of the storage array A_val will hold the value $A_{ij}$ for $1 \leq i \leq m$, $1 \leq j \leq n$. The string A_type = ‘dense_by_columns’ should be specified.

Sparse co-ordinate storage format: Only the nonzero entries of the matrices are stored. For the $l$-th entry, $1 \leq l \leq ne$, of $A$, its row index i, column index j and value $A_{ij}$, $1 \leq i \leq m$, $1 \leq j \leq n$, are stored as the $l$-th components of the integer arrays A_row and A_col and real array A_val, respectively, while the number of nonzeros is recorded as A_ne = $ne$. The string A_type = ‘coordinate’should be specified.

Sparse row-wise storage format: Again only the nonzero entries are stored, but this time they are ordered so that those in row i appear directly before those in row i+1. For the i-th row of $A$ the i-th component of the integer array A_ptr holds the position of the first entry in this row, while A_ptr(m+1) holds the total number of entries plus one. The column indices j, $1 \leq j \leq n$, and values $A_{ij}$ of the nonzero entries in the i-th row are stored in components l = A_ptr(i), $\ldots$, A_ptr(i+1)-1, $1 \leq i \leq m$, of the integer array A_col, and real array A_val, respectively. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string A_type = ‘sparse_by_rows’ should be specified.

Sparse column-wise storage format: Once again only the nonzero entries are stored, but this time they are ordered so that those in column j appear directly before those in column j+1. For the j-th column of $A$ the j-th component of the integer array A_ptr holds the position of the first entry in this column, while A_ptr(n+1) holds the total number of entries plus one. The row indices i, $1 \leq i \leq m$, and values $A_{ij}$ of the nonzero entries in the j-th columnsare stored in components l = A_ptr(j), $\ldots$, A_ptr(j+1)-1, $1 \leq j \leq n$, of the integer array A_row, and real array A_val, respectively. As before, for sparse matrices, this scheme almost always requires less storage than the co-ordinate format. The string A_type = ‘sparse_by_columns’ should be specified.

symmetric storage#

The symmetric $n$ by $n$ matrix $H$ may also be presented and stored in a variety of formats. But crucially symmetry is exploited by only storing values from the lower triangular part (i.e, those entries that lie on or below the leading diagonal).

Dense storage format: The matrix $H$ is stored as a compact dense matrix by rows, that is, the values of the entries of each row in turn are stored in order within an appropriate real one-dimensional array. Since $H$ is symmetric, only the lower triangular part (that is the part $H_{ij}$ for $1 \leq j \leq i \leq n$) need be held. In this case the lower triangle should be stored by rows, that is component $(i-1) * i / 2 + j$ of the storage array H_val will hold the value $H_{ij}$ (and, by symmetry, $H_{ji}$) for $1 \leq j \leq i \leq n$. The string H_type = ‘dense’ should be specified.

Sparse co-ordinate storage format: Only the nonzero entries of the matrices are stored. For the $l$-th entry, $1 \leq l \leq ne$, of $H$, its row index i, column index j and value $H_{ij}$, $1 \leq j \leq i \leq n$, are stored as the $l$-th components of the integer arrays H_row and H_col and real array H_val, respectively, while the number of nonzeros is recorded as H_ne = $ne$. Note that only the entries in the lower triangle should be stored. The string H_type = ‘coordinate’ should be specified.

Sparse row-wise storage format: Again only the nonzero entries are stored, but this time they are ordered so that those in row i appear directly before those in row i+1. For the i-th row of $H$ the i-th component of the integer array H_ptr holds the position of the first entry in this row, while H_ptr(n+1) holds the total number of entries plus one. The column indices j, $1 \leq j \leq i$, and values $H_{ij}$ of the entries in the i-th row are stored in components l = H_ptr(i), …, H_ptr(i+1)-1 of the integer array H_col, and real array H_val, respectively. Note that as before only the entries in the lower triangle should be stored. For sparse matrices, this scheme almost always requires less storage than its predecessor. The string H_type = ‘sparse_by_rows’ should be specified.

Diagonal storage format: If $H$ is diagonal (i.e., $H_{ij} = 0$ for all $1 \leq i \neq j \leq n$) only the diagonals entries $H_{ii}$, $1 \leq i \leq n$ need be stored, and the first n components of the array H_val may be used for the purpose. The string H_type = ‘diagonal’ should be specified.

Multiples of the identity storage format: If $H$ is a multiple of the identity matrix, (i.e., $H = \alpha I$ where $I$ is the n by n identity matrix and $\alpha$ is a scalar), it suffices to store $\alpha$ as the first component of H_val. The string H_type = ‘scaled_identity’ should be specified.

The identity matrix format: If $H$ is the identity matrix, no values need be stored. The string H_type = ‘identity’ should be specified.

The zero matrix format: The same is true if $H$ is the zero matrix, but now the string H_type = ‘zero’ or ‘none’ should be specified.

introduction to function calls#

To solve a given problem, functions from the cqp package must be called in the following order:

cqp_initialize - provide default control parameters and set up initial data structures
cqp_read_specfile (optional) - override control values by reading replacement values from a file
cqp_import - set up problem data structures and fixed values
cqp_reset_control (optional) - possibly change control parameters if a sequence of problems are being solved
solve the problem by calling one of
- cqp_solve_qp - solve the quadratic program
- cqp_solve_sldqp - solve the shifted least-distance problem
cqp_information (optional) - recover information about the solution and solution process
cqp_terminate - deallocate data structures

See the examples section for illustrations of use.

parametric real type T#

Below, the symbol T refers to a parametric real type that may be Float32 (single precision) or Float64 (double precision).

callable functions#

    function cqp_initialize(T, data, control, status)

Set default control values and initialize private data

Parameters:

data

holds private internal data

control

is a structure containing control information (see cqp_control_type)

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):

0
The import was successful.

    function cqp_read_specfile(T, control, specfile)

Read the content of a specification file, and assign values associated with given keywords to the corresponding control parameters. An in-depth discussion of specification files is available, and a detailed list of keywords with associated default values is provided in $GALAHAD/src/cqp/CQP.template. See also Table 2.1 in the Fortran documentation provided in $GALAHAD/doc/cqp.pdf for a list of how these keywords relate to the components of the control structure.

Parameters:

control	is a structure containing control information (see cqp_control_type)
specfile	is a one-dimensional array of type Vararg{Cchar} that must give the name of the specification file

function cqp_import(T, control, data, status, n, m, H_type, H_ne, H_row,
                    H_col, H_ptr, A_type, A_ne, A_row, A_col, A_ptr)

Import problem data into internal storage prior to solution.

Parameters:

control	is a structure whose members provide control parameters for the remaining procedures (see cqp_control_type)
data	holds private internal data
status	is a scalar variable of type Int32 that gives the exit status from the package. Possible values are: 0 The import was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 or m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -23 An entry from the strict upper triangle of $H$ has been specified.
n	is a scalar variable of type Int32 that holds the number of variables.
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
H_type	is a one-dimensional array of type Vararg{Cchar} that specifies the symmetric storage scheme used for the Hessian, $H$. It should be one of ‘coordinate’, ‘sparse_by_rows’, ‘dense’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’, the latter pair if $H=0$; lower or upper case variants are allowed.
H_ne	is a scalar variable of type Int32 that holds the number of entries in the lower triangular part of $H$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.
H_row	is a one-dimensional array of size H_ne and type Int32 that holds the row indices of the lower triangular part of $H$ in the sparse co-ordinate storage scheme. It need not be set for any of the other three schemes, and in this case can be C_NULL.
H_col	is a one-dimensional array of size H_ne and type Int32 that holds the column indices of the lower triangular part of $H$ in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense, diagonal or (scaled) identity storage schemes are used, and in this case can be C_NULL.
H_ptr	is a one-dimensional array of size n+1 and type Int32 that holds the starting position of each row of the lower triangular part of $H$, as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be C_NULL.
A_type	is a one-dimensional array of type Vararg{Cchar} that specifies the unsymmetric storage scheme used for the constraint Jacobian, $A$. It should be one of ‘coordinate’, ‘sparse_by_rows’ or ‘dense; lower or upper case variants are allowed.
A_ne	is a scalar variable of type Int32 that holds the number of entries in $A$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes.
A_row	is a one-dimensional array of size A_ne and type Int32 that holds the row indices of $A$ in the sparse co-ordinate storage scheme. It need not be set for any of the other schemes, and in this case can be C_NULL.
A_col	is a one-dimensional array of size A_ne and type Int32 that holds the column indices of $A$ in either the sparse co-ordinate, or the sparse row-wise storage scheme. It need not be set when the dense or diagonal storage schemes are used, and in this case can be C_NULL.
A_ptr	is a one-dimensional array of size n+1 and type Int32 that holds the starting position of each row of $A$, as well as the total number of entries, in the sparse row-wise storage scheme. It need not be set when the other schemes are used, and in this case can be C_NULL.

function cqp_reset_control(T, control, data, status)

Reset control parameters after import if required.

Parameters:

control

is a structure whose members provide control parameters for the remaining procedures (see cqp_control_type)

data

holds private internal data

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are:

0 The import was successful.

function cqp_solve_qp(T, data, status, n, m, h_ne, H_val, g, f, a_ne, A_val,
                      c_l, c_u, x_l, x_u, x, c, y, z, x_stat, c_stat)

Solve the quadratic program when the Hessian $H$ is available.

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -5 The simple-bound constraints are inconsistent. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem. -23 An entry from the strict upper triangle of $H$ has been specified.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
h_ne	is a scalar variable of type Int32 that holds the number of entries in the lower triangular part of the Hessian matrix $H$.
H_val	is a one-dimensional array of size h_ne and type T that holds the values of the entries of the lower triangular part of the Hessian matrix $H$ in any of the available storage schemes.
g	is a one-dimensional array of size n and type T that holds the linear term $g$ of the objective function. The j-th component of `g`, j = 1, … , n, contains $g_j$.
f	is a scalar of type T that holds the constant term $f$ of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix $A$.
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix $A$ in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds $c^l$ on the constraints $A x$. The i-th component of `c_l`, i = 1, … , m, contains $c^l_i$.
c_u	is a one-dimensional array of size m and type T that holds the upper bounds $c^l$ on the constraints $A x$. The i-th component of `c_u`, i = 1, … , m, contains $c^u_i$.
x_l	is a one-dimensional array of size n and type T that holds the lower bounds $x^l$ on the variables $x$. The j-th component of `x_l`, j = 1, … , n, contains $x^l_j$.
x_u	is a one-dimensional array of size n and type T that holds the upper bounds $x^l$ on the variables $x$. The j-th component of `x_u`, j = 1, … , n, contains $x^l_j$.
x	is a one-dimensional array of size n and type T that holds the values $x$ of the optimization variables. The j-th component of `x`, j = 1, … , n, contains $x_j$.
c	is a one-dimensional array of size m and type T that holds the residual $c(x)$. The i-th component of `c`, i = 1, … , m, contains $c_i(x)$.
y	is a one-dimensional array of size n and type T that holds the values $y$ of the Lagrange multipliers for the general linear constraints. The i-th component of `y`, i = 1, … , m, contains $y_i$.
z	is a one-dimensional array of size n and type T that holds the values $z$ of the dual variables. The j-th component of `z`, j = 1, … , n, contains $z_j$.
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable $x_j$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^Tx$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

function cqp_solve_sldqp(T, data, status, n, m, w, x0, g, f, a_ne, A_val,
                         c_l, c_u, x_l, x_u, x, c, y, z, x_stat, c_stat)

Solve the shifted least-distance quadratic program

Parameters:

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -5 The simple-bound constraints are inconsistent. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
w	is a one-dimensional array of size n and type T that holds the values of the weights $w$.
x0	is a one-dimensional array of size n and type T that holds the values of the shifts $x^0$.
g	is a one-dimensional array of size n and type T that holds the linear term $g$ of the objective function. The j-th component of `g`, j = 1, … , n, contains $g_j$.
f	is a scalar of type T that holds the constant term $f$ of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix $A$.
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix $A$ in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds $c^l$ on the constraints $A x$. The i-th component of `c_l`, i = 1, … , m, contains $c^l_i$.
c_u	is a one-dimensional array of size m and type T that holds the upper bounds $c^l$ on the constraints $A x$. The i-th component of `c_u`, i = 1, … , m, contains $c^u_i$.
x_l	is a one-dimensional array of size n and type T that holds the lower bounds $x^l$ on the variables $x$. The j-th component of `x_l`, j = 1, … , n, contains $x^l_j$.
x_u	is a one-dimensional array of size n and type T that holds the upper bounds $x^l$ on the variables $x$. The j-th component of `x_u`, j = 1, … , n, contains $x^l_j$.
x	is a one-dimensional array of size n and type T that holds the values $x$ of the optimization variables. The j-th component of `x`, j = 1, … , n, contains $x_j$.
c	is a one-dimensional array of size m and type T that holds the residual $c(x)$. The i-th component of `c`, i = 1, … , m, contains $c_i(x)$.
y	is a one-dimensional array of size n and type T that holds the values $y$ of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, i = 1, … , m, contains $y_i$.
z	is a one-dimensional array of size n and type T that holds the values $z$ of the dual variables. The j-th component of `z`, j = 1, … , n, contains $z_j$.
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable $x_j$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value $a_i^T x$ most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

function cqp_information(T, data, inform, status)

Provides output information

Parameters:

data

holds private internal data

inform

is a structure containing output information (see cqp_inform_type)

status

is a scalar variable of type Int32 that gives the exit status from the package. Possible values are (currently):

0 The values were recorded successfully

function cqp_terminate(T, data, control, inform)

Deallocate all internal private storage

Parameters:

data	holds private internal data
control	is a structure containing control information (see cqp_control_type)
inform	is a structure containing output information (see cqp_inform_type)

available structures#

cqp_control_type structure#

    struct cqp_control_type{T}
      f_indexing::Bool
      error::Int32
      out::Int32
      print_level::Int32
      start_print::Int32
      stop_print::Int32
      maxit::Int32
      infeas_max::Int32
      muzero_fixed::Int32
      restore_problem::Int32
      indicator_type::Int32
      arc::Int32
      series_order::Int32
      sif_file_device::Int32
      qplib_file_device::Int32
      infinity::T
      stop_abs_p::T
      stop_rel_p::T
      stop_abs_d::T
      stop_rel_d::T
      stop_abs_c::T
      stop_rel_c::T
      perturb_h::T
      prfeas::T
      dufeas::T
      muzero::T
      tau::T
      gamma_c::T
      gamma_f::T
      reduce_infeas::T
      obj_unbounded::T
      potential_unbounded::T
      identical_bounds_tol::T
      mu_pounce::T
      indicator_tol_p::T
      indicator_tol_pd::T
      indicator_tol_tapia::T
      cpu_time_limit::T
      clock_time_limit::T
      remove_dependencies::Bool
      treat_zero_bounds_as_general::Bool
      treat_separable_as_general::Bool
      just_feasible::Bool
      getdua::Bool
      puiseux::Bool
      every_order::Bool
      feasol::Bool
      balance_initial_complentarity::Bool
      crossover::Bool
      space_critical::Bool
      deallocate_error_fatal::Bool
      generate_sif_file::Bool
      generate_qplib_file::Bool
      sif_file_name::NTuple{31,Cchar}
      qplib_file_name::NTuple{31,Cchar}
      prefix::NTuple{31,Cchar}
      fdc_control::fdc_control_type{T}
      sbls_control::sbls_control_type{T}
      fit_control::fit_control_type
      roots_control::roots_control_type{T}
      cro_control::cro_control_type{T}

detailed documentation#

control derived type as a Julia structure

components#

Bool f_indexing

use C or Fortran sparse matrix indexing

Int32 error

error and warning diagnostics occur on stream error

Int32 out

general output occurs on stream out

Int32 print_level

the level of output required is specified by print_level

$\leq$ 0 gives no output,
= 1 gives a one-line summary for every iteration,
= 2 gives a summary of the inner iteration for each iteration,
$\geq$ 3 gives increasingly verbose (debugging) output

Int32 start_print

any printing will start on this iteration

Int32 stop_print

any printing will stop on this iteration

Int32 maxit

at most maxit inner iterations are allowed

Int32 infeas_max

the number of iterations for which the overall infeasibility of the problem is not reduced by at least a factor .reduce_infeas before the problem is flagged as infeasible (see reduce_infeas)

Int32 muzero_fixed

the initial value of the barrier parameter will not be changed for the first muzero_fixed iterations

Int32 restore_problem

indicate whether and how much of the input problem should be restored on output. Possible values are

0 nothing restored
1 scalar and vector parameters
2 all parameters

Int32 indicator_type

specifies the type of indicator function used. Possible values are

1 primal indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_p_tol
2 primal-dual indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_tol_pd \* size of corresponding multiplier
3 primal-dual indicator: a constraint is active if and only if the distance to its nearest bound $\leq$.indicator_tol_tapia \* distance to same bound at previous iteration

Int32 arc

which residual trajectory should be used to aim from the current iterate to the solution. Possible values are

1 the Zhang linear residual trajectory
2 the Zhao-Sun quadratic residual trajectory
3 the Zhang arc ultimately switching to the Zhao-Sun residual trajectory
4 the mixed linear-quadratic residual trajectory
5 the Zhang arc ultimately switching to the mixed linear-quadratic residual trajectory

Int32 series_order

the order of (Taylor/Puiseux) series to fit to the path data

Int32 sif_file_device

specifies the unit number to write generated SIF file describing the current problem

Int32 qplib_file_device

specifies the unit number to write generated QPLIB file describing the current problem

T infinity

any bound larger than infinity in modulus will be regarded as infinite

T stop_abs_p

the required absolute and relative accuracies for the primal infeasibility

T stop_rel_p

see stop_abs_p

T stop_abs_d

the required absolute and relative accuracies for the dual infeasibility

T stop_rel_d

see stop_abs_d

T stop_abs_c

the required absolute and relative accuracies for the complementarity

T stop_rel_c

see stop_abs_c

T perturb_h

.perturb_h will be added to the Hessian

T prfeas

initial primal variables will not be closer than .prfeas from their bounds

T dufeas

initial dual variables will not be closer than .dufeas from their bounds

T muzero

the initial value of the barrier parameter. If muzero is not positive, it will be reset to an appropriate value

T tau

the weight attached to primal-dual infeasibility compared to complementa when assessing step acceptance

T gamma_c

individual complementarities will not be allowed to be smaller than gamma_c times the average value

T gamma_f

the average complementarity will not be allowed to be smaller than gamma_f times the primal/dual infeasibility

T reduce_infeas

if the overall infeasibility of the problem is not reduced by at least a factor .reduce_infeas over .infeas_max iterations, the problem is flagged as infeasible (see infeas_max)

T obj_unbounded

if the objective function value is smaller than obj_unbounded, it will be flagged as unbounded from below.

T potential_unbounded

if W=0 and the potential function value is smaller than .potential_unbounded $\ast$ number of one-sided bounds, the analytic center will be flagged as unbounded

T identical_bounds_tol

any pair of constraint bounds $(c_l,c_u)$ or $(x_l,x_u)$ that are closer than .identical_bounds_tol will be reset to the average of their values

T mu_pounce

start terminal extrapolation when mu reaches mu_pounce

T indicator_tol_p

if .indicator_type = 1, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_p_tol

T indicator_tol_pd

if .indicator_type = 2, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_tol_pd \* size of corresponding multiplier

T indicator_tol_tapia

if .indicator_type = 3, a constraint/bound will be deemed to be active if and only if the distance to its nearest bound $\leq$.indicator_tol_tapia \* distance to same bound at previous iteration

T cpu_time_limit

the maximum CPU time allowed (-ve means infinite)

T clock_time_limit

the maximum elapsed clock time allowed (-ve means infinite)

Bool remove_dependencies

the equality constraints will be preprocessed to remove any linear dependencies if true

Bool treat_zero_bounds_as_general

any problem bound with the value zero will be treated as if it were a general value if true

Bool treat_separable_as_general

if .just_feasible is true, the algorithm will stop as soon as a feasible point is found. Otherwise, the optimal solution to the problem will be found

Bool just_feasible

if .treat_separable_as_general, is true, any separability in the problem structure will be ignored

Bool getdua

if .getdua, is true, advanced initial values are obtained for the dual variables

Bool puiseux

decide between Puiseux and Taylor series approximations to the arc

Bool every_order

try every order of series up to series_order?

Bool feasol

if .feasol is true, the final solution obtained will be perturbed so that variables close to their bounds are moved onto these bounds

Bool balance_initial_complentarity

if .balance_initial_complentarity is true, the initial complemetarity is required to be balanced

Bool crossover

if .crossover is true, cross over the solution to one defined by linearly-independent constraints if possible

Bool space_critical

if .space_critical true, every effort will be made to use as little space as possible. This may result in longer computation time

Bool deallocate_error_fatal

if .deallocate_error_fatal is true, any array/pointer deallocation error will terminate execution. Otherwise, computation will continue

Bool generate_sif_file

if .generate_sif_file is .true. if a SIF file describing the current problem is to be generated

Bool generate_qplib_file

if .generate_qplib_file is .true. if a QPLIB file describing the current problem is to be generated

NTuple{31,Cchar} sif_file_name

name of generated SIF file containing input problem

NTuple{31,Cchar} qplib_file_name

name of generated QPLIB file containing input problem

NTuple{31,Cchar} prefix

all output lines will be prefixed by .prefix(2:LEN(TRIM(.prefix))-1) where .prefix contains the required string enclosed in quotes, e.g. “string” or ‘string’

struct fdc_control_type fdc_control

control parameters for FDC

struct sbls_control_type sbls_control

control parameters for SBLS

struct fit_control_type fit_control

control parameters for FIT

struct roots_control_type roots_control

control parameters for ROOTS

struct cro_control_type cro_control

control parameters for CRO

cqp_time_type structure#

struct cqp_time_type{T}
      total::T
      preprocess::T
      find_dependent::T
      analyse::T
      factorize::T
      solve::T
      clock_total::T
      clock_preprocess::T
      clock_find_dependent::T
      clock_analyse::T
      clock_factorize::T
      clock_solve::T

detailed documentation#

time derived type as a Julia structure

components#

T total

the total CPU time spent in the package

T preprocess

the CPU time spent preprocessing the problem

T find_dependent

the CPU time spent detecting linear dependencies

T analyse

the CPU time spent analysing the required matrices prior to factorization

T factorize

the CPU time spent factorizing the required matrices

T solve

the CPU time spent computing the search direction

T clock_total

the total clock time spent in the package

T clock_preprocess

the clock time spent preprocessing the problem

T clock_find_dependent

the clock time spent detecting linear dependencies

T clock_analyse

the clock time spent analysing the required matrices prior to factorization

T clock_factorize

the clock time spent factorizing the required matrices

T clock_solve

the clock time spent computing the search direction

cqp_inform_type structure#

    struct cqp_inform_type{T}
      status::Int32
      alloc_status::Int32
      bad_alloc::NTuple{81,Cchar}
      iter::Int32
      factorization_status::Int32
      factorization_integer::Int64
      factorization_real::Int64
      nfacts::Int32
      nbacts::Int32
      threads::Int32
      obj::T
      primal_infeasibility::T
      dual_infeasibility::T
      complementary_slackness::T
      init_primal_infeasibility::T
      init_dual_infeasibility::T
      init_complementary_slackness::T
      potential::T
      non_negligible_pivot::T
      feasible::Bool
      checkpointsIter::NTuple{16,Int32}
      checkpointsTime::NTuple{16,T}
      time::cqp_time_type{T}
      fdc_inform::fdc_inform_type{T}
      sbls_inform::sbls_inform_type{T}
      fit_inform::fit_inform_type
      roots_inform::roots_inform_type
      cro_inform::cro_inform_type{T}
      rpd_inform::rpd_inform_type

detailed documentation#

inform derived type as a Julia structure

components#

Int32 status

return status. See CQP_solve for details.

Int32 alloc_status

the status of the last attempted allocation/deallocation.

NTuple{81,Cchar} bad_alloc

the name of the array for which an allocation/deallocation error occurred.

Int32 iter

the total number of iterations required

Int32 factorization_status

the return status from the factorization

Int64 factorization_integer

the total integer workspace required for the factorization

Int64 factorization_real

the total real workspace required for the factorization

Int32 nfacts

the total number of factorizations performed

Int32 nbacts

the total number of “wasted” function evaluations during the linesearch

Int32 threads

the number of threads used

T obj

the value of the objective function at the best estimate of the solution determined by CQP_solve

T primal_infeasibility

the value of the primal infeasibility

T dual_infeasibility

the value of the dual infeasibility

T complementary_slackness

the value of the complementary slackness

T init_primal_infeasibility

these values at the initial point (needed bg GALAHAD_CCQP)

T init_dual_infeasibility

see init_primal_infeasibility

T init_complementary_slackness

see init_primal_infeasibility

T potential

the value of the logarithmic potential function sum -log(distance to constraint boundary)

T non_negligible_pivot

the smallest pivot which was not judged to be zero when detecting linear dependent constraints

Bool feasible

is the returned “solution” feasible?

Int32 checkpointsIter[16]

checkpoints(i) records the iteration at which the criticality measures first fall below $10^{-i-1}$, i = 0, …, 15 (-1 means not achieved)

T checkpointsTime[16]

see checkpointsIter

struct cqp_time_type time

timings (see above)

struct fdc_inform_type fdc_inform

inform parameters for FDC

struct sbls_inform_type sbls_inform

inform parameters for SBLS

struct fit_inform_type fit_inform

return information from FIT

struct roots_inform_type roots_inform

return information from ROOTS

struct cro_inform_type cro_inform

inform parameters for CRO

struct rpd_inform_type rpd_inform

inform parameters for RPD

Index

example calls#

This is an example of how to use the package to solve a given convex quadratic program; the code is available in $GALAHAD/src/cqp/Julia/test_cqp.jl . A variety of supported Hessian and constraint matrix storage formats are shown.

# test_cqp.jl
# Simple code to test the Julia interface to CQP

using GALAHAD
using Test
using Printf
using Accessors

function test_cqp(::Type{T}) where T
  # Derived types
  data = Ref{Ptr{Cvoid}}()
  control = Ref{cqp_control_type{T}}()
  inform = Ref{cqp_inform_type{T}}()

  # Set problem data
  n = 3 # dimension
  m = 2 # number of general constraints
  H_ne = 3 # Hesssian elements
  H_row = Cint[1, 2, 3]  # row indices, NB lower triangle
  H_col = Cint[1, 2, 3]  # column indices, NB lower triangle
  H_ptr = Cint[1, 2, 3, 4]  # row pointers
  H_val = T[1.0, 1.0, 1.0]  # values
  g = T[0.0, 2.0, 0.0]  # linear term in the objective
  f = 1.0  # constant term in the objective
  A_ne = 4 # Jacobian elements
  A_row = Cint[1, 1, 2, 2]  # row indices
  A_col = Cint[1, 2, 2, 3]  # column indices
  A_ptr = Cint[1, 3, 5]  # row pointers
  A_val = T[2.0, 1.0, 1.0, 1.0]  # values
  c_l = T[1.0, 2.0]  # constraint lower bound
  c_u = T[2.0, 2.0]  # constraint upper bound
  x_l = T[-1.0, -Inf, -Inf]  # variable lower bound
  x_u = T[1.0, Inf, 2.0]  # variable upper bound

  # Set output storage
  c = zeros(T, m) # constraint values
  x_stat = zeros(Cint, n) # variable status
  c_stat = zeros(Cint, m) # constraint status
  st = ' '
  status = Ref{Cint}()

  @printf(" Fortran sparse matrix indexing\n\n")
  @printf(" basic tests of qp storage formats\n\n")

  for d in 1:7
    # Initialize CQP
    cqp_initialize(T, data, control, status)

    # Set user-defined control options
    @reset control[].f_indexing = true # Fortran sparse matrix indexing

    # Start from 0
    x = T[0.0, 0.0, 0.0]
    y = T[0.0, 0.0]
    z = T[0.0, 0.0, 0.0]

    # sparse co-ordinate storage
    if d == 1
      st = 'C'
      cqp_import(T, control, data, status, n, m,
                 "coordinate", H_ne, H_row, H_col, C_NULL,
                 "coordinate", A_ne, A_row, A_col, C_NULL)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # sparse by rows
    if d == 2
      st = 'R'
      cqp_import(T, control, data, status, n, m,
                 "sparse_by_rows", H_ne, C_NULL, H_col, H_ptr,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end
    # dense
    if d == 3
      st = 'D'
      H_dense_ne = 6 # number of elements of H
      A_dense_ne = 6 # number of elements of A
      H_dense = T[1.0, 0.0, 1.0, 0.0, 0.0, 1.0]
      A_dense = T[2.0, 1.0, 0.0, 0.0, 1.0, 1.0]

      cqp_import(T, control, data, status, n, m,
                 "dense", H_ne, C_NULL, C_NULL, C_NULL,
                 "dense", A_ne, C_NULL, C_NULL, C_NULL)

      cqp_solve_qp(T, data, status, n, m, H_dense_ne, H_dense, g, f,
                   A_dense_ne, A_dense, c_l, c_u, x_l, x_u,
                   x, c, y, z, x_stat, c_stat)
    end

    # diagonal
    if d == 4
      st = 'L'
      cqp_import(T, control, data, status, n, m,
                 "diagonal", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # scaled identity
    if d == 5
      st = 'S'
      cqp_import(T, control, data, status, n, m,
                 "scaled_identity", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # identity
    if d == 6
      st = 'I'
      cqp_import(T, control, data, status, n, m,
                 "identity", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    # zero
    if d == 7
      st = 'Z'
      cqp_import(T, control, data, status, n, m,
                 "zero", H_ne, C_NULL, C_NULL, C_NULL,
                 "sparse_by_rows", A_ne, C_NULL, A_col, A_ptr)

      cqp_solve_qp(T, data, status, n, m, H_ne, H_val, g, f,
                   A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                   x_stat, c_stat)
    end

    cqp_information(T, data, inform, status)

    if inform[].status == 0
      @printf("%c:%6i iterations. Optimal objective value = %5.2f status = %1i\n",
              st, inform[].iter, inform[].obj, inform[].status)
    else
      @printf("%c: CQP_solve exit status = %1i\n", st, inform[].status)
    end

    # @printf("x: ")
    # for i = 1:n
    #   @printf("%f ", x[i])
    # end
    # @printf("\n")
    # @printf("gradient: ")
    # for i = 1:n
    #   @printf("%f ", g[i])
    # end
    # @printf("\n")

    # Delete internal workspace
    cqp_terminate(T, data, control, inform)
  end

  # test shifted least-distance interface
  for d in 1:1
    # Initialize CQP
    cqp_initialize(T, data, control, status)

    # Set user-defined control options
    @reset control[].f_indexing = true # Fortran sparse matrix indexing

    # Start from 0
    x = T[0.0, 0.0, 0.0]
    y = T[0.0, 0.0]
    z = T[0.0, 0.0, 0.0]

    # Set shifted least-distance data

    w = T[1.0, 1.0, 1.0]
    x_0 = T[0.0, 0.0, 0.0]

    # sparse co-ordinate storage
    if d == 1
      st = 'W'
      cqp_import(T, control, data, status, n, m,
                 "shifted_least_distance", H_ne, C_NULL, C_NULL, C_NULL,
                 "coordinate", A_ne, A_row, A_col, C_NULL)

      cqp_solve_sldqp(T, data, status, n, m, w, x_0, g, f,
                      A_ne, A_val, c_l, c_u, x_l, x_u, x, c, y, z,
                      x_stat, c_stat)
    end

    cqp_information(T, data, inform, status)

    if inform[].status == 0
      @printf("%c:%6i iterations. Optimal objective value = %5.2f status = %1i\n",
              st, inform[].iter, inform[].obj, inform[].status)
    else
      @printf("%c: CQP_solve exit status = %1i\n", st, inform[].status)
    end

    # @printf("x: ")
    # for i = 1:n
    #   @printf("%f ", x[i])
    # end
    # @printf("\n")
    # @printf("gradient: ")
    # for i = 1:n
    #   @printf("%f ", g[i])
    # end
    # @printf("\n")

    # Delete internal workspace
    cqp_terminate(T, data, control, inform)
  end

  return 0
end

@testset "CQP" begin
  @test test_cqp(Float32) == 0
  @test test_cqp(Float64) == 0
end

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful. -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -5 The simple-bound constraints are inconsistent. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem. -23 An entry from the strict upper triangle of \(H\) has been specified.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
h_ne	is a scalar variable of type Int32 that holds the number of entries in the lower triangular part of the Hessian matrix \(H\).
H_val	is a one-dimensional array of size h_ne and type T that holds the values of the entries of the lower triangular part of the Hessian matrix \(H\) in any of the available storage schemes.
g	is a one-dimensional array of size n and type T that holds the linear term \(g\) of the objective function. The j-th component of `g`, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_l`, i = 1, … , m, contains \(c^l_i\).
c_u	is a one-dimensional array of size m and type T that holds the upper bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_u`, i = 1, … , m, contains \(c^u_i\).
x_l	is a one-dimensional array of size n and type T that holds the lower bounds \(x^l\) on the variables \(x\). The j-th component of `x_l`, j = 1, … , n, contains \(x^l_j\).
x_u	is a one-dimensional array of size n and type T that holds the upper bounds \(x^l\) on the variables \(x\). The j-th component of `x_u`, j = 1, … , n, contains \(x^l_j\).
x	is a one-dimensional array of size n and type T that holds the values \(x\) of the optimization variables. The j-th component of `x`, j = 1, … , n, contains \(x_j\).
c	is a one-dimensional array of size m and type T that holds the residual \(c(x)\). The i-th component of `c`, i = 1, … , m, contains \(c_i(x)\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the general linear constraints. The i-th component of `y`, i = 1, … , m, contains \(y_i\).
z	is a one-dimensional array of size n and type T that holds the values \(z\) of the dual variables. The j-th component of `z`, j = 1, … , n, contains \(z_j\).
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value \(a_i^Tx\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.

data	holds private internal data
status	is a scalar variable of type Int32 that gives the entry and exit status from the package. Possible exit values are: 0 The run was successful -1 An allocation error occurred. A message indicating the offending array is written on unit control.error, and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -2 A deallocation error occurred. A message indicating the offending array is written on unit control.error and the returned allocation status and a string containing the name of the offending array are held in inform.alloc_status and inform.bad_alloc respectively. -3 The restrictions n > 0 and m > 0 or requirement that a type contains its relevant string ‘dense’, ‘coordinate’, ‘sparse_by_rows’, ‘diagonal’, ‘scaled_identity’, ‘identity’, ‘zero’ or ‘none’ has been violated. -5 The simple-bound constraints are inconsistent. -7 The constraints appear to have no feasible point. -9 The analysis phase of the factorization failed; the return status from the factorization package is given in the component inform.factor_status -10 The factorization failed; the return status from the factorization package is given in the component inform.factor_status. -11 The solution of a set of linear equations using factors from the factorization package failed; the return status from the factorization package is given in the component inform.factor_status. -16 The problem is so ill-conditioned that further progress is impossible. -17 The step is too small to make further impact. -18 Too many iterations have been performed. This may happen if control.maxit is too small, but may also be symptomatic of a badly scaled problem. -19 The CPU time limit has been reached. This may happen if control.cpu_time_limit is too small, but may also be symptomatic of a badly scaled problem.
n	is a scalar variable of type Int32 that holds the number of variables
m	is a scalar variable of type Int32 that holds the number of general linear constraints.
w	is a one-dimensional array of size n and type T that holds the values of the weights \(w\).
x0	is a one-dimensional array of size n and type T that holds the values of the shifts \(x^0\).
g	is a one-dimensional array of size n and type T that holds the linear term \(g\) of the objective function. The j-th component of `g`, j = 1, … , n, contains \(g_j\).
f	is a scalar of type T that holds the constant term \(f\) of the objective function.
a_ne	is a scalar variable of type Int32 that holds the number of entries in the constraint Jacobian matrix \(A\).
A_val	is a one-dimensional array of size a_ne and type T that holds the values of the entries of the constraint Jacobian matrix \(A\) in any of the available storage schemes.
c_l	is a one-dimensional array of size m and type T that holds the lower bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_l`, i = 1, … , m, contains \(c^l_i\).
c_u	is a one-dimensional array of size m and type T that holds the upper bounds \(c^l\) on the constraints \(A x\). The i-th component of `c_u`, i = 1, … , m, contains \(c^u_i\).
x_l	is a one-dimensional array of size n and type T that holds the lower bounds \(x^l\) on the variables \(x\). The j-th component of `x_l`, j = 1, … , n, contains \(x^l_j\).
x_u	is a one-dimensional array of size n and type T that holds the upper bounds \(x^l\) on the variables \(x\). The j-th component of `x_u`, j = 1, … , n, contains \(x^l_j\).
x	is a one-dimensional array of size n and type T that holds the values \(x\) of the optimization variables. The j-th component of `x`, j = 1, … , n, contains \(x_j\).
c	is a one-dimensional array of size m and type T that holds the residual \(c(x)\). The i-th component of `c`, i = 1, … , m, contains \(c_i(x)\).
y	is a one-dimensional array of size n and type T that holds the values \(y\) of the Lagrange multipliers for the general linear constraints. The j-th component of `y`, i = 1, … , m, contains \(y_i\).
z	is a one-dimensional array of size n and type T that holds the values \(z\) of the dual variables. The j-th component of `z`, j = 1, … , n, contains \(z_j\).
x_stat	is a one-dimensional array of size n and type Int32 that gives the optimal status of the problem variables. If x_stat(j) is negative, the variable \(x_j\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.
c_stat	is a one-dimensional array of size m and type Int32 that gives the optimal status of the general linear constraints. If c_stat(i) is negative, the constraint value \(a_i^T x\) most likely lies on its lower bound, if it is positive, it lies on its upper bound, and if it is zero, it lies between its bounds.